On Friday, November 14, 2025 at 8:30:29 PM UTC-7 Alan Grayson wrote:
On Friday, November 14, 2025 at 7:42:23 PM UTC-7 Russell Standish wrote: T(u,v) = uᵀMv where M is the matrix representation of T, and ᵀ is the transpose operator. This is too succinct for me to understand your explanation. AG *Is the above how a tensor is generally evaluated? How would it be evaluated if* *T has three independent variables, u,v,w? AG* *If u and v are modeled as row matrices, then u transposed is a colum vector, and* *the total result is a real number. But I don't think it's easy to show that this is the* *same value obtained by modeling the tensor as a linear function of u and v. Offhand.* *do you have a link for showing the equivalence? Further, in the case of T(u), how is * *the result a real number when using matrix notation? It looks like a vector when u* *is modeled as a column vector. OTOH, I don't think we can model u as a row vector* *to do the **calculation in this situation (or can we?). AG * On Fri, Nov 14, 2025 at 05:12:46PM -0800, Alan Grayson wrote: > > > On Thursday, November 13, 2025 at 11:14:13 PM UTC-7 Russell Standish wrote: > > On Sat, Nov 08, 2025 at 05:25:16AM -0800, Alan Grayson wrote: > > In some treatments of tensors, they're described as linear maps. So, in > GR, if > > we have a linear map described as a 4x4 matrix of real numbers, which > operates > > on a 4-vector described as a column matrix with entries (ct, x, y, z), > which > > transforms to another 4-vector, what must be added in this description to > claim > > that the linear transformation satisfies the definition of a tensor? TY, > AG > > The operation of matrix multiplication is linear over the vectors on > the right hand side. > > Is that what you're missing? > > > Suppose T is a tensor which operates on vector v in some vector space. Then T > (v) > maps to a real number and is linear in v. If we change coordinates, v's > coordinates will > change but v remains the same, so T is invariant wrt coordinate > transformations. > Now let's look at the situation using matrix representation and assume we're > dealing > with a 4 dimensional spacetime manifold. Then T can be represented by a 4x4 > matrix, > and v can be represented as a column matrix with entries ct, x, y, z. When T > operates > on v, we get another column vector, and not a real constant! What am I doing > wrong? > Doesn't every tensor map to a real number? Also, if T is a function of two > vectors, T(u,v), > then using matrix notation, how is T evaluated to get a real number as the > result? Do > we model u as a column matrix and v as a row matrix? Thanks for your time. AG > > > ---------------------------------------------------------------------------- > > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders [email protected] > http://www.hpcoders.com.au > ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/d4496344-074c-4df8-9703-5dd029accf82n%40googlegroups.com.
